A novel analytical formulation of the Axelrod model
Lucía Pedraza, Sebastián Pinto, Juan Pablo Pinasco, Pablo Balenzuela
AA novel analytical formulation of the Axelrod model
Luc´ıa Pedraza , Sebasti´an Pinto , Juan Pablo Pinasco , and Pablo Balenzuela Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires. Av.Cantilos/n, Pabell´on 1, Ciudad Universitaria, 1428, Buenos Aires, Argentina. Instituto de F´ısica de Buenos Aires (IFIBA), CONICET. Av.Cantilo s/n, Pabell´on 1, Ciudad Universitaria,1428, Buenos Aires, Argentina. Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires andIMAS UBA-CONICET, Av. Cantilo s/n, Pabell´on 1, Ciudad Universitaria, 1428, Buenos Aires, Argentina.
June 30, 2020
Abstract
The Axelrod model of cultural dissemination has been widely studied in the field of statisticalmechanics. The traditional version of this agent-based model is to assign a cultural vector of F components to each agent, where each component can take one of Q cultural trait. In this work, weintroduce a novel set of mean field master equations to describe the model for F = 2 and F = 3 incomplete graphs where all indirect interactions are explicitly calculated. We find that the transitionbetween different macroscopic states is driven by initial conditions (set by parameter Q ) and the sizeof the system N , who measures the balance between linear and cubic terms in master equations. Wealso find that this analytical approach fully agrees with simulations where the system does not breakup during the dynamics and a scaling relation related to missing links reestablishes the agreementwhen this happens. The Axelrod model [1] has been proposed to explain the phenomenon of polarization in a a society inwhich individuals always are looking for a consensus in their opinions. This model is based on twowell-established mechanisms: Social influence, through which people become more similar when theyinteract; and homophily, which is the tendency of individuals to interact preferentially with similarones. The mathematical description of this model consists on describing individuals as agents, each onedescribed by a vector of F components called cultural features which at the same time can take oneof Q integer values called cultural traits. When two agents interact with a probability proportional totheir shared features, one of them copies a feature from the other one. Despite the simplicity of themodel and beyond its original formulation, several variants are still being proposed in order to improvethe modeling of polarization, such as the emergence of new topics [4], a layered organization of socialinteractions [2], the inclusion of peer-pressure [10], and the formation of opinion-based groups [9].Which is interesting from a statistical physics point of view is that this model shows a non-equilibriumphase transition from a monocultural to a multicultural state. This phase transition takes place byvarying the number of cultural traits Q for a given fixed F . If the number of cultural traits is low, theprobability of interaction is high, leading the system to a monocultural state. On the other hand, if Q is high, the mentioned probability is low and the system evolves to a stationary multicultural stateafter a few interactions. This phase transition was studied in several topologies such as one-dimensionalsystems [6, 8], lattices [3], complex [5] and complete networks [11].Although the Axelrod model is usually studied through numerical simulations, different analyticalapproaches were developed based on stochastic equations [3, 12] or deterministic systems [14, 13], thatdescribe the evolution of the density of bonds with a given similarity and provide some insights aboutthe origin of the phase-transition. However, these approaches rely on several approximations needed todiscard high-order terms and make equations analytical tractable. Moreover, the more fully descriptiveworks are necessarily only devoted to the simplest case F = 2 [13, 7].In this work, we develop a new set of mean field equations on complete networks that exactly describethe average behavior of the Axelrod model in the F = 2 case and provide a full description of the similarity1 a r X i v : . [ phy s i c s . s o c - ph ] J un istribution dynamics for larger F . These equations are based on a novel formulation of the model interms of similarity vectors among agents. This formulation naturally takes into account correlationsamong cultural states and simplifies the model due to the parameter Q is only involved in setting theinitial condition. We show that the F = 2 case reduces to a trivial dynamical behavior, while the case F = 3 shows a competition between linear and cubic terms mediated by the size of the system N . Inthis last case, the analytical approach shows a fully agreement with simulations when N increases atfixed Q (below the transition point), but fails during the transition where the mean-field hypothesis donot hold. However, the agreement is rapidly recovered by a scaling factor related to missing links due tothe system fragmentation during the dynamics. The Axelrod model [1] describes each agent by a vector of F components which can take one of Q integervalues. That vector represents a set of cultural features associated to a given individual, and the differentvalues a component adopts represent different cultural traits related to a given feature. The model startsby creating random cultural states for the agents. In this work, the initial state is set by assigningwith equal probability one of Q integer values to each cultural feature. Once the initial condition isestablished, the dynamics of the system is based on a pairwise interaction mechanism, which relies ontwo fundamental hypothesis: • Homophily: The probability of interaction between two topological connected individuals is pro-portional to their cultural similarity, that is, the number of features they share. More specifically,two agents interact with probability n/F , where n is the number of shared features. If n = F , theagents do not interact. • Social Influence: After each interaction, the agents become more similar. It means that one of theagents copies a feature from the other which they previously did not share.The system evolves until there are no active links in the system, i.e., all connected agents either do notshare any feature or share F cultural features.This model shows a non-equilibrium phase transition from a monocultural to a multicultural stateby varying the value of Q for a fixed F . When Q < Q c , the probability that two agents can interactsince the initial state is high, so all agents end with the same cultural vector. On the other hand, when Q > Q c , the probability of interaction at the initial state is low and the final state shows a coexistence ofregions with different cultural states. The transition point and the order of the phase-transition dependson the values of F and on the topology of the underlying contact network. In this work, we study themodel on a complete network, where a given agent can interact with any other agent in the system. . Here we postulate an alternative formulation of Axelrod model based on link’s dynamics insteadof agent’s dynamics. Given than similarity between agents plays a main role in the dynamics of themodel, we introduce this new formulation in terms of similarity vectors. This framework will allow usto write closed mean field master equations for describing the dynamics of the model, as we shown innext sections. Figure 1 shows how the description based on individual cultural vectors is seen in termsof similarity vectors associated to any pairs of agents. This formulation implies to describe the systemof N agents, with their original N cultural vectors, in terms of N ( N − similarity vectors.A similarity vector between two agents is a F-dimensional binary vector which has an X in the placewhere they share a cultural feature and 0 otherwise. Both formulation are identical (see Appendix fordetails) if they fulfill the following relationships: Given a state of the similarity vector between agent i and j ( i − j ), if some feature adopts the value of X , then the respective feature in the similarity vectors i − k and j − k must be equal (0 or X ) for all k closing the triangle i − j − k . Any change in a similarityvector must fulfill this condition. (A full description of the dynamics is sketched in the Appendix).The formulation in terms of similarity vectors shows that the parameter Q only plays a role in settingthe initial state: A similarity vector of length F and n X -values appears at the initial state with aprobability: P ( n | F, Q ) = (cid:18) Fn (cid:19) ( 1 Q ) n (1 − Q ) F − n (1)2nce the initial state is set, the absolute value of Q is meaningless from the point of view of the dynamics.This independence of Q is already present in the mean-field approach given in [3], but it seems to be lostin most of the Axelrod literature. Figure 1:
Axelrod model described in terms of similarity vectors.
Cultural vectors (left figure)of F components and Q integer values per feature associated with each agent are replaced by binarysimilarity vectors (right figure) of F components associated to each pair of agents. The formulation in terms of similarity vectors allows to derive mean-field equations for the similaritydistribution in terms of the density of states. We introduce here the analytical approach for the cases F = 2 and F = 3. F = 2 In this case, a given link can be in four states: P = [0 , P a = [ X, P b = [0 , X ] and P = [ X, X ].These states can not coexist in a given trio of agents due to the closure relationship detailed in previoussection. Moreover, given the homophily driven pairwise interaction, a change in the cultural state of oneagent due to a direct interaction with his partner produces not only an update in the similarity of thecurrent link but also indirect updates in the similarities of all the other pairs of agents which involve theformer one.Figure 2 shows two examples of direct and indirect link updates that give rise to some terms of theequations. Bold lines represent those links where a direct change can take place, and correspond to states P a or P b because direct changes do not take place if similarity is zero ( P ) or one ( P ). If we consider allpossible combinations and direct changes, these lead to the following equations for the similarity vectorstates, where direct changes produce linear terms and indirect changes are reflected in cubic terms inthe equations: dP dt = ( N − (cid:16) − P a P b P − P a P b P + P a P b P + P a P b P (cid:17) dP a dt = − P a N − (cid:16) − P a P − P P a P b + P P a P b + P a P (cid:17) dP b dt = − P b N − (cid:16) − P b P − P P a P b + P P a P b + P b P (cid:17) dP dt = P b + P a N − (cid:16) − P a P − P b P + P a P + P b P (cid:17) , where the ( N −
2) factor represents the amount of indirect links that might change, considering a completegraph. As we can see in these last equations, the indirect changes effect is canceled and only the directdynamic remains. By calling P to the density of links in states P a or P b , and considering that both3 and b type are equally probably, we finally obtain a set of equations for F = 2 that can be explicitlysolved: dP dt = 0 = ⇒ P ( t ) = P ( t = 0) dP dt = − P ⇒ P ( t ) = P ( t = 0) e − t dP dt = P ⇒ P ( t ) = 1 − P ( t = 0) − P ( t = 0) e − t (2)These equations fulfill the normalization constraint P + P + P = 1, and the initial condition is givenby Eq.(1).Figure 3 shows the comparison between equations (2) and the average of the similarity distribution ina complete graph Axelrod model. In this figure, we can see the fully agreement between analytical andsimulation results: Equations (2) correctly predict that P remains constant during the whole dynamicsand equal to its initial value (1 − /Q ) , while P and P exponentially decay. On the other hand, wedo not observe a dependence on N once the time scale is adjusted as dt = M where M = N ( N − . P (0,0) P (0,X)P (X,0) P (0,X) P (0,X)P (X,X) d P a dt ∼− P a d P dt ∼− P a P b P d P dt ∼ P a d P b dt ∼ P a P b P Direct changes Indirect changes P (0,X) P (0,0)P (X,0) P (0,0) P (0,0)P (X,X) d P a dt ∼− P a d P b dt ∼− P a P b P d P dt ∼ P a d P dt ∼ P a P b P Figure 2:
Example of dynamics of vector similarity states.
Two examples of links updates whichinvolves direct and indirect interactions. Note that direct changes lead to linear terms in Eq.(2), andthese direct changes imply indirect ones, pointed out by dashed lines, which at the same time lead tocubic terms The 1 / /F for F = 2 and one cultural feature shared).The conservation of P provides a picture of the Axelrod dynamics in an average sense, which can bevisualized as a rearrangement of non-zero similarity links in connected components that are also cliquesin the final state. Finally, we want to remark that the dependency on Q (as stated above) is only presentin the initial similarity distribution ( P i ( t = 0)) and these parameter plays no role during the dynamics.4igure 3: Analytical prediction for F = 2 . Time evolution of similarity distribution for F = 2and Q = 2 (panel (a)), and P at the final state (panel (b)). Dots belong to simulations while linesare analytical predictions. N = 256, but no dependence on N was observed. Time is measured in M interactions, where M = N ( N − the number of pair of agents in the system. F = 3 Following the same approach detailed above, we write down the dynamical equations for the similarityvectors when F = 3 (See Appendix for details). In this case, we obtain: dP dt = ( N − (cid:16) P − P P P (cid:17) dP dt = − P N − (cid:16) − P − P P + P P + 6 P P P (cid:17) dP dt = P − P N − (cid:16) P + 6 P P − P P − P P P (cid:17) dP dt = 2 P N − (cid:16) − P P + P P (cid:17) (3)Due to the normalization condition P + P + P + P = 1, this equation system is actually a three-dimensional one. As in the previous case, the initial condition is given by Eq.(1).The equation system (3) has a set of fixed points at P = P = 0 which corresponds to the case whenthere are no active links in the system. This condition does not impose any constraint on P and P .Therefore, the stationary state is fully characterized by specifying the value of P at the final state, P f ,which in principle can be any value between 0 and 1. On the other hand, P f is determined through thenormalization condition P f = 1 − P f . The eigenvalues of the linearized equations are { , − / , − / , } ,which confirm that the infinite fixed points are stables (see Appendix for details). The equation system(3) has also an isolated fixed point which explicitly depends on N , but it is an unfeasible solution dueto some P i fall out of the range [0 ,
1] and therefore is discarded for future analysis.To which of the final values of P the system converges depends on the parameters Q and N , as theleft panel of figure 4 shows. Here we can observe that for small values of N (roughly N < P f = 0 to P f = 1 as Q increases, while P f shows an abrupt jump from0 to 1 when N is roughly greater than 100. This jump takes place at a critical value of Q which scaleslinearly with N , following the relationship Q c /N (cid:39) . P f depends on Q and N , these parameters play two well-separated roles. In one hand, Q determines the initial conditions of the system according to Eq. (1), while on the other hand, N weightsthe coupling of the cubic terms with the linear ones. Before going on the analysis of the entire system,it is interesting to look at the case of large N : Here, we could neglect the linear terms respect to thecubic ones, leading to a new equation system without linear terms. This approximation presents a setof stable fixed points with a predominant attractive component which fulfills the following relationships: P = (1 − √ P )( √ P ) P = (1 − √ P ) √ P and P = (1 − √ P ) (the stability analysis of thisapproximation can be found in the Appendix).Given that P f = P f = 0, in panel (b) of figure 4 we analyze the trajectories of the system in the P - P plane for different values of N at fixed Q . Here, the straight line of slope − P = (1 − √ P ) , is satisfied by P and P at the initial condition given by the Axelrod model(Eq.( 1)). This means that, at t = 0, the cubic terms of equations (3) become equal to zero and thedynamics is driven only by the linear terms (See Appendix for details).In panel (b), the four trajectories sketched ( Q = 10 and N = 2; 50; 75; 250) display how the balancebetween linear and cubic terms rules the dynamics: When N is small, the linear term is dominant andthe system evolves to the closest fixed point in the straight line, as is shown in the case N = 2 wherecubic terms are absent. However, when N is large (for instance, N = 250), the linear terms drives thedynamics at t = 0, but once the system leaves the dashed gray line, these are negligible with respect tothe cubic ones, and the system is driven by the stability of that curve. Then, the trajectory goes to thestability points in the straight line, but following the dashed gray line as can be seen in the figure (redtrajectory in panel (b)). In this example, for intermediates values of N , the dynamics is driven by thecompetition between the linear and cubic terms. For larger values of Q , the system starts from regionsof higher values of P and either jumps to the closest stable states on the straight line (which also meanshigh values of P f ) for small values of N , or moves following the cubic-stable manifold until reaching thestable state at low values of P f for large values of N .The comparison between the analytical approach and simulations can be observed in Figure 5. Incontrast to F = 2, the F = 3 case shows a dependence on N and the matching between analyticalequations and simulations improves when N increases for finite Q . When this happen, P decays to zeroleading to a monocultural state in the thermodynamic limit for finite Q ( Q/N → Q/N ∼ . Phase diagrams of the analytical system.
Stationary solution P f as a function of N and Q (left panel), and phase diagram for different N with same initial condition ( Q = 10) (right panel). Inthe last one, the dashed line is the set of initial conditions, the solid line is the set of fixed points, andarrows point out time direction. In terms of the similarity distribution, the phase transition of the Axelrod model (the passage from amonocultural state to a multicultural state when Q increases) corresponds to a change in P from zeroto a non-zero value at the final state ( P f ). Top panel of figure (6) shows this transition together withthe predicted value of the analytical approach as a function of Q/N . This scaling was in part suggestedby the phase diagram in panel (a) of figure (4) . As top panel of Figure 6 shows, the analytical approachdiffers respect to the Axelrod model during the transition, but it matches the simulations for low valuesand high values of
Q/N . In this figure, it is clearer that the analytical system shows a critical valueequal to
Q/N ∼ . P f between the Axelrod model and the analytical approach below Q/N ∼ . a)(c) (b)(d) Figure 5:
Phase diagrams or Time evolution for F = 3 , N = 512 and Q = 12 (panels (a) and(b)) and (panels (c) and (d)). Dots belong to simulations while lines belong to the analyticalapproach. Panels (a) and (c) show the time evolution of the similarity distribution, while panels (b) and(d) show the same information as phase diagrams.group, they act as two independent fragments. Once these clusters appear, the Axelrod model hasno mechanisms to join them again. For instance, bottom panel of figure (6) shows the multiplicity offragments for N = 1024. As can be observed, the system begins made up by a unique fragment but endedup fragmented, which is seen as an increment in the multiplicity. When the system is fragmented, thehypothesis of a mean-field approach fails. Panel (b) also suggests an explanation of why the analyticalapproach shows a transition in P : the critical value ( Q c /N ∼ .
4) coincides with the value of
Q/N atwhich the system is already fragmented at the initial state.On the other hand, since every link inside a fragment has similarity equal to 1 at the final state(otherwise the system will keep evolving), a given value of P f in Axelrod simulations corresponds to thenumber of links between fragments (which ended up with similarity equal to 0), normalized by the totalnumber of links in the system. This means that difference between these simulations and theoreticalvalues should correspond to the missing links due to fragmentation during the dynamics. In order to testthis hypothesis, we modify the value of P f in theoretical calculations by adding the contributions dueto the inter-fragments missing links from the knowledge of the fragment distribution at the final state.When doing this, the modified value of P f (dashed yellow curve) matches exactly the simulations, as canbe seen in upper panel of Figure 6. An interesting result is that by modifying P with the missed links ofthe final fragments, it produces a good agreement with simulations for the whole dynamical trajectories.We can call ˆ P i to the analytical values of P i corrected by the missing links. If P inter is the fraction ofzero-similarity links due to the missing inter-fragment links, this reads as:ˆ P = P inter + (1 − P inter ) P ˆ P = (1 − P inter ) P ˆ P = (1 − P inter ) P ˆ P = (1 − P inter ) P . (4)Figure 7 shows that this modification produces an excellent approximation of the trajectory in the7 - P plane although is not so good when they are shown as function of time. The idea behind the re-scaling proposed is that, if we would know the fragment distribution of the stationary state, we removedthe inter-fragment links (that we know will end up with zero similarity) at t = 0 as if they would notcontribute during the whole dynamics. Then, the analytical equations refers to the similarity distributioninside fragments, and the matching with simulations are recovered when combining with inter-fragmentlinks of the stationary state through Eq.(4).Figure 6: Axelrod and analytical transition in terms of P f . Top right panel shows the transitionfor different N . Bottom right panel show the multiplicity (number) of fragments at the initial and finalstate of the Axelrod model for N = 1024. It can be seen that the analytical transition corresponds to aninitial fragmented system, while the difference between the Axelrod model and the mean-field approachcorresponds to a fragmented system at the final state. In this work, we present a novel mean-field approach of the Axelrod model based on similarity vectors oncomplete networks for F = 2 and F = 3. In our analytical approach, once F is set, the system dependson two parameters that play well-separated roles: On one hand, the parameter Q (together with F ) setthe initial condition of the similarity distribution, while the system size N couples the linear terms withcubic ones in master equations.Using this approach, we were able to exactly reproduce the dynamics of the similarity distribution for F = 2 for all value of Q , and correctly predict that, in this case, the dynamics does not depend on N oncethe time step is set to 1 /M , where M is the number of pairs of agents in the system. In the case of F = 3,our approach reproduces simulations while the system is made up by a unique fragment, condition that issatisfied in the thermodynamic limit for fixed Q ( Q/N → Axelrod and analytical transition in terms of P f . Re-scaling dynamic as time function(left) and trajectory (right) for F = 3, N = 512 and Q = 128.can break up into groups of agents with orthogonal cultural states and the model has non mechanismto join them again. When this happens, an irreversible amount of links with similarity zero ( P inter )is created, the mean-field hypothesis does not hold, and the analytical approach fails. However, by are-scaling of the similarity distribution which involves P inter , the trajectories of simulations in phasediagrams can be recovered from the analytical equations.Specifically for the analytical system at F = 3, we found a transition for large N at Q/N ∼ . P f jumps from 0 to 1. This values can be respectivelyidentified with the monocultural and multicultural phases of the Axelrod model. We do not providehere a theoretical explanation of why this transition lies around 0 .
4, but we think that this can be foundby exploring the competition between the eigenvalues and eigenvectors of the linear system and thenon-linear system (made up by only the cubic terms).Our work is not the first in proposing master-equations for the Axelrod model and follows pastattempts like [3, 13]. In all cases, the idea is to write a master-equation for the similarity distribution P m , which in general looks like: dP m dt = F − (cid:88) r =1 rF P k (cid:104) δ m,r +1 − δ m,r + ( g − F (cid:88) n =0 ( P n W ( r ) n,m − P m W ( r ) m,n ) (cid:105) where g is the coordination number of the underlying network and W ( r ) n,m are transition rates that takeinto account the probability of an indirect change for n to m due to a direct change with r shared culturalfeatures.The main difference between analytical approaches lies in the calculation of W ( r ) n,m . Although usefulinsights about the Axelrod transition can be extracted for both [3] and [13], in these approaches correla-tions among adjacent links are neglected in order to make equations analytical tractable, leading to aninaccurate description of the dynamics of the system. In particular, in [13] the transition rates involvea parameter λ (interpreted as the conditional probability that two agents, i and k , share a feature thatis simultaneously not shared with a third one j ), that is approximated by λ = ( Q − − , involving theparameter Q in the dynamics which we show can be exactly decoupled from it. In our approach, corre-lations among agents are explicitly taken into account by writing the possible combination of similarityvectors.Another important difference in our approach is that when an agent i copy a feature from j changingits cultural state, we also update the N − i and any other agent k (cid:54) = j , nomatter if the pair i − k is topologically connected or not. Although in this work we consider a completenetwork where every pair of agents is connected, it is an important feature that must be considered ifthis approach is extrapolated to other network topologies. In [3, 13], only connected links are takeninto account, which implies the presence of the coordination number g in the general form of the masterequation given above.Finally, although we restrict our analysis to the cases F = 2 and F = 3, our approach can be seen9s an algorithm to figure out all possible combinations of similarity vectors that can be translated intomaster-equations for larger values of F . References [1] Robert Axelrod. The dissemination of culture: A model with local convergence and global polar-ization.
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A The Axelrod model and the similarity vector approach
To reproduce the same dynamics than the Axelrod model, the dynamical rules must be rewritten interms of the similarity vectors. Figure 8 shows an example of the effects of changing the value of onesimilarity feature. The most important fact is that, given three agents (which in this case means threesimilarity vectors), if we look at a specific similarity feature there is one banned state: To take the valueof X in two vectors and 0 in the last one. For instance, consider figure 8 again: When the similarityfeature of i − j changes to X , if the feature i − k doesn’t change this would imply that agents i and j agree in that cultural feature, j and k do the same, but i and k not, which is a contradiction.Summarizing, the dynamical rules of the Axelrod model in terms of similarity vectors are the following:10 Take a similarity vector of two connected agents, i and j . With a probability proportional to thenumber of X in their vector (which is the same as the homophily in the Axelrod model), change arandom feature with value 0 to X . • Now, suppose that the change in the similarity vector comes from implicitly changing the value ofa cultural feature in the state of i . Then, for all k (cid:54) = i, j , set the value of the similarity feature i − k equal to the respective feature in j − k .Finally, given the similarity vector i − j , if a given feature adopts the value of X , then the respectivefeature in the similarity vectors i − k and j − k must be equal for all i, j, k . x x xi jk i jk Direct changeIndirect change
Figure 8:
Dynamical behaviour in terms of similarity vectors.
Suppose that one of the similarityfeatures between agents i and j is changed from 0 to X by a direct change, and that it occurs whenagent i copies a cultural feature from j . Since only the cultural state of i changes, the similarity featurebetween j and k remains constant. Then, the respective feature between i and k must necessarily changeand adopt the same value that the similarity feature between j and k in order to avoid the banned statedescribed in the main text.Figure 9 shows the relative size of the biggest fragment at the final state S max /N , in the Axelrodmodel in terms of both cultural and similarity vectors. S max /N is the usual observable to characterizethe Axelrod transition. As figure shows, there are no significant differences in choosing one representationor the other. B Derivation of F = 3 case master equations For F = 3, let P be the proportion of the total of the links that are the similarity vector [0 , , P a , P b , P c the three vectors for a link with one feature in common, P a , P b , P c with two and P thevector for a link that joins two equal states. Similarly to the F = 2 we write all the feasibly terns wherea direct change made an indirect one. Figure 10 show two examples for direct and indirect changes. Theequation for the dynamic of these states is: dP dt = ( N − (cid:16) − P ( P c P c + P b P b + P a P a ) + 3 P a P b P c (cid:17) dP a dt = − P a N − (cid:16) − P a P − P a P b P c + P a P b P c ++ P P b P b + P P c P c − P a ( P c + P b )2 (cid:17) dP a dt = ( P b + P c )6 − P a N − (cid:16) − P a P a P − P a ( P b P c + P b P c )++ P b P c ( P b + P c )2 + P ( P b + P c ) (cid:17) dP dt =2 P a + P b + P c N − (cid:16) − P ( P a + P b + P c ) + P a P b P c ++ P a P c P b + P b P a P c (cid:17) = 2 F = 3 F = 5 Figure 9: Axelrod model described by both cultural (full lines) and similarity vectors (triangles) on acomplete network for N = 1024 agents and different values of F . Inset shows same results for N = 256.This figure shows that both representations are equivalent.Finally, we assume that different states with the same amount of features in common remain sym-metric, that is P a = P b = P c = P and P a = P b = P c = P , obtaining equations: dP dt = ( N − (cid:16) P − P P P (cid:17) dP dt = − P N − (cid:16) − P − P P + P P + 6 P P P (cid:17) dP dt = P − P N − (cid:16) P + 6 P P − P P − P P P (cid:17) dP dt = 2 P N − (cid:16) − P P + P P (cid:17) C Fixed points and stability analysis of F = 3 case master equa-tions As mentioned in the main text, in addition to the set of fixed points when P = P = 0, this system hasthe following isolated fixed point: P = − ± (cid:114) − cP = 3 ∓ (cid:114) − cP = − ± (cid:114) − cP = 32 ± (cid:114) − c where c = ( N − /
27. However, these point is an unfeasible one due to P = − P and therefore one ofthem is necessarily negative number. 12igure 10: Example of dynamics of vector similarity states.
Two examples of links updates whichinvolves direct and indirect interactions. Note that direct changes lead to linear terms in Eq.(3), andthese direct changes imply indirect ones, pointed out by dashed lines, which at the same time lead tocubic terms The 1 / / /F and 2 /F for F = 3 and one or two cultural feature shared) and the probability that the featurechanged is the illustrated one (i.e., 1 / P = P = 0, we obtain the matrix: L = − −
00 0 whose eigenvalues and eigenvectors are: λ = − / v = (0 , , − , λ = − / v = (0 , − / , − / , λ = 0 v = (0 , , ,
1) y v = (1 , , , D Stability of the non-linear part
By taking only the non-linear terms, we can calculate the first order matrix and evaluate it around initialconditions. The linearized matrix is read as: D = − P P P − P P − P P P P − P − P P + 6 P P + P P P + 2 P P − P − P P − P P − P + 12 P P + 3 P − P P − P P P − P P + P P P − P We numerically calculate the eigenvalues of this matrix at the initial condition. Figure 11 shows thesevalues as function of the initial P . As we can see in this figure, the most dominant eigenvalue hasnegative sign, leading that the initial conditions are also a stable manifold.13 .0 0.2 0.4 0.6 0.8 1.0 P E i g e n v a l u e s Q Figure 11:
Eigenvalues for the linearized system of the non-linear terms.
Left figure as functionof P = (1 − /Q ) , right figure as function of Q . E Null non-linear terms at initial condition
We can rewrite Eq.(3) by defining A = P − P P and B = P − P P : dP dt = ( n − P AdP dt = − P n − P ( B − A ) dP dt = P − P n − P ( A − B ) dP dt = 2 P n − P B At a given Q , the initial similarity distribution is: P = (cid:16) − q (cid:17) P = 3 (cid:16) − q (cid:17) qP = 3 (cid:16) − q (cid:17)(cid:16) q (cid:17) P = (cid:16) q (cid:17) In this case both A and B are null: A = 9 (cid:16) − q (cid:17) (cid:16) q (cid:17) − (cid:16) − q (cid:17) (cid:104) (cid:16) − q (cid:17)(cid:16) q (cid:17) (cid:105) = 9 (cid:16) − q (cid:17) (cid:16) q (cid:17) − (cid:16) − q (cid:17) (cid:16) q (cid:17) = 0 B = 9 (cid:16) − q (cid:17) (cid:16) q (cid:17) − (cid:104) (cid:16) − q (cid:17) q (cid:105)(cid:16) q (cid:17)3